The \(O(N)\)  models count among the most paradigmatic systems in the theory of critical phenomena and were with great success applied to address universal characteristics of an amazingly broad variety of physical situations [1]}, [2]}. Even though the physically most relevant cases correspond to integer number of order-parameter components \(N\)  and integer spatial dimensionality \(d\) , it has proven extremely fruitful to consider these quantities formally as continuous parameters, leading to the development of theoretical approaches such as the \((4-\epsilon )\) -expansion, \((2+\epsilon )\) -expansion, or the \(1/N\) -expansion, where one accesses the most relevant range of parameters (\(d=3\)  in particular) by expanding around an analytically soluble point in the \((d, N)\) -plane. It is also worthwhile observing that there has recently been certain interest (both experimental and theoretical) in engineering situations, where the effective dimensionality of the system would not coincide with the physical dimensionality and, in particular, might take a fractional value (see e.g. Refs. [3]}, [4]}, [5]}). Also note that mathematically rigorous meaning can be provided for continuous range of \(N\)  [6]}.
